Tomographic reconstruction is an ill-conditioned estimation problem where noise present in measurement data can be greatly magnified by the image reconstruction process. Thus, accurate reconstruction techniques must carefully balance noise in the reconstruction (which arises from noise in the measurements) with prior information that can be used to mitigate noise (typically using notions of image smoothness—perhaps with edge-preservation). So-called model-based reconstruction approaches achieve this balance using an explicit model for the measurement noise. These approaches, also called statistical reconstruction methods, tend to provide better image quality over nonstatistical methods (e.g., filtered-backprojection (FBP)) that do not leverage a noise model. These approaches can also permit the reduction of radiation dose without loss of imaging performance. As such, the accuracy of the measurement model, including the specific noise model, can influence image quality. Generally, one expects that increasing fidelity of the measurement model will improve image quality and/or dose utilization.
The detection process in tomography varies across modalities (e.g., PET, SPECT, CT, and cone-beam CT (CBCT)) and specific devices and hardware vendors. Moreover, exact mathematical modeling of the detection process can be difficult, and approximations are often made. For example, in CBCT using flat-panel detectors (FPDs), x-ray photons are generated according to a process well-modeled by Poisson noise, but are detected by an indirect process whereby x-ray photons are converted in a phosphor or scintillator to visible light photons (another statistical process in which a randomly distributed number of optical photons are produced for each interacting x-ray), which in turn spread within the scintillator (causing image blur and noise correlation) and are converted to electrical charge in a semiconductor, amplified, and quantized in an analog-to-digital convertor (yet another statistical process in terms of the amount of noise added by the electronics). The resulting model is complex, neither purely Gaussian nor Poisson. The cascade of statistical processes stemming from the initially Poisson-distribute fluence of incident photons includes: Bernoulli selection (interaction of x-ray in the converter); Poisson+Excess (conversion to optical photons); Bernoulli selection (conversion to electrical charge); and Gaussian (electronics noise). Although the resulting statistical process can be difficult to write down in a closed mathematical form, a cascaded systems analysis of signal and noise transfer characteristics in the imaging chain has shown to provide a very accurate model of first-order (mean) and second-order (variance) statistics. One commonly used alternative is to approximate the noise model using an appropriately parameterized Gaussian (or Poisson) model. In many systems, this assumption also leads to a very accurate model.
One assumption on the noise model that appears universal in current model-based reconstruction is that noise is independent across measurements. That is, there are no correlations between measurements. While this may be a reasonable assumption, for example, in CT detectors where blur and cross-talk between detector elements is minimal, this assumption is generally not true and is a very coarse approximation to FPDs. In FPDs, which are a prevalent form of detector in CBCT, such correlations in the signal and noise can be substantial due to the physics of detection. (During the conversion of x-ray photons to visible light photons, many visible photons are generated from a single x-ray, and these visible photon spread spatially in the detector creating a “patch” of correlated signal.)
Accordingly, there is a need in the art for a method of tomographic reconstruction that models the covariance of measurements in the forward model to improve image quality, increase spatial resolution and enhance detectability.